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Theorem pwunss 4011
 Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)
Assertion
Ref Expression
pwunss (𝒫 A ∪ 𝒫 B) ⊆ 𝒫 (AB)

Proof of Theorem pwunss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ssun 3116 . . 3 ((xA xB) → x ⊆ (AB))
2 elun 3078 . . . 4 (x (𝒫 A ∪ 𝒫 B) ↔ (x 𝒫 A x 𝒫 B))
3 vex 2554 . . . . . 6 x V
43elpw 3357 . . . . 5 (x 𝒫 AxA)
53elpw 3357 . . . . 5 (x 𝒫 BxB)
64, 5orbi12i 680 . . . 4 ((x 𝒫 A x 𝒫 B) ↔ (xA xB))
72, 6bitri 173 . . 3 (x (𝒫 A ∪ 𝒫 B) ↔ (xA xB))
83elpw 3357 . . 3 (x 𝒫 (AB) ↔ x ⊆ (AB))
91, 7, 83imtr4i 190 . 2 (x (𝒫 A ∪ 𝒫 B) → x 𝒫 (AB))
109ssriv 2943 1 (𝒫 A ∪ 𝒫 B) ⊆ 𝒫 (AB)
 Colors of variables: wff set class Syntax hints:   ∨ wo 628   ∈ wcel 1390   ∪ cun 2909   ⊆ wss 2911  𝒫 cpw 3351 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353 This theorem is referenced by:  pwundifss  4013  pwunim  4014
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